3.1126 \(\int \frac{(A+B x) (b x+c x^2)^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=255 \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac{d^2 (B d-A e) (c d-b e)^2}{7 e^6 (d+e x)^7}+\frac{c (-A c e-2 b B e+5 B c d)}{3 e^6 (d+e x)^3}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{6 e^6 (d+e x)^6}-\frac{B c^2}{2 e^6 (d+e x)^2} \]

[Out]

(d^2*(B*d - A*e)*(c*d - b*e)^2)/(7*e^6*(d + e*x)^7) - (d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e
)))/(6*e^6*(d + e*x)^6) - (A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))/
(5*e^6*(d + e*x)^5) + (2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))/(4*e^6*(d + e*x)^4) + (c*
(5*B*c*d - 2*b*B*e - A*c*e))/(3*e^6*(d + e*x)^3) - (B*c^2)/(2*e^6*(d + e*x)^2)

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Rubi [A]  time = 0.223629, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac{d^2 (B d-A e) (c d-b e)^2}{7 e^6 (d+e x)^7}+\frac{c (-A c e-2 b B e+5 B c d)}{3 e^6 (d+e x)^3}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{6 e^6 (d+e x)^6}-\frac{B c^2}{2 e^6 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^8,x]

[Out]

(d^2*(B*d - A*e)*(c*d - b*e)^2)/(7*e^6*(d + e*x)^7) - (d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e
)))/(6*e^6*(d + e*x)^6) - (A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))/
(5*e^6*(d + e*x)^5) + (2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))/(4*e^6*(d + e*x)^4) + (c*
(5*B*c*d - 2*b*B*e - A*c*e))/(3*e^6*(d + e*x)^3) - (B*c^2)/(2*e^6*(d + e*x)^2)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^8} \, dx &=\int \left (-\frac{d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^8}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^7}+\frac{A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^5 (d+e x)^6}+\frac{-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{e^5 (d+e x)^5}+\frac{c (-5 B c d+2 b B e+A c e)}{e^5 (d+e x)^4}+\frac{B c^2}{e^5 (d+e x)^3}\right ) \, dx\\ &=\frac{d^2 (B d-A e) (c d-b e)^2}{7 e^6 (d+e x)^7}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{6 e^6 (d+e x)^6}-\frac{A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{5 e^6 (d+e x)^5}+\frac{2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{4 e^6 (d+e x)^4}+\frac{c (5 B c d-2 b B e-A c e)}{3 e^6 (d+e x)^3}-\frac{B c^2}{2 e^6 (d+e x)^2}\\ \end{align*}

Mathematica [A]  time = 0.114121, size = 260, normalized size = 1.02 \[ -\frac{2 A e \left (2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 b c e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+2 c^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (3 b^2 e^2 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+8 b c e \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+10 c^2 \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )\right )}{420 e^6 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^8,x]

[Out]

-(2*A*e*(2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b*c*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 2*c^
2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + B*(3*b^2*e^2*(d^3 + 7*d^2*e*x + 21*d*e^2*x
^2 + 35*e^3*x^3) + 8*b*c*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 10*c^2*(d^5 + 7*d^
4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)))/(420*e^6*(d + e*x)^7)

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Maple [A]  time = 0.006, size = 307, normalized size = 1.2 \begin{align*}{\frac{d \left ( 2\,A{b}^{2}{e}^{3}-6\,Abcd{e}^{2}+4\,A{c}^{2}{d}^{2}e-3\,{b}^{2}Bd{e}^{2}+8\,B{d}^{2}bce-5\,B{c}^{2}{d}^{3} \right ) }{6\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{{d}^{2} \left ( A{b}^{2}{e}^{3}-2\,Abcd{e}^{2}+A{c}^{2}{d}^{2}e-{b}^{2}Bd{e}^{2}+2\,B{d}^{2}bce-B{c}^{2}{d}^{3} \right ) }{7\,{e}^{6} \left ( ex+d \right ) ^{7}}}-{\frac{2\,Abc{e}^{2}-4\,A{c}^{2}de+B{e}^{2}{b}^{2}-8\,Bdbce+10\,B{c}^{2}{d}^{2}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{c \left ( Ace+2\,bBe-5\,Bcd \right ) }{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{B{c}^{2}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{A{b}^{2}{e}^{3}-6\,Abcd{e}^{2}+6\,A{c}^{2}{d}^{2}e-3\,{b}^{2}Bd{e}^{2}+12\,B{d}^{2}bce-10\,B{c}^{2}{d}^{3}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^8,x)

[Out]

1/6*d*(2*A*b^2*e^3-6*A*b*c*d*e^2+4*A*c^2*d^2*e-3*B*b^2*d*e^2+8*B*b*c*d^2*e-5*B*c^2*d^3)/e^6/(e*x+d)^6-1/7*d^2*
(A*b^2*e^3-2*A*b*c*d*e^2+A*c^2*d^2*e-B*b^2*d*e^2+2*B*b*c*d^2*e-B*c^2*d^3)/e^6/(e*x+d)^7-1/4*(2*A*b*c*e^2-4*A*c
^2*d*e+B*b^2*e^2-8*B*b*c*d*e+10*B*c^2*d^2)/e^6/(e*x+d)^4-1/3*c*(A*c*e+2*B*b*e-5*B*c*d)/e^6/(e*x+d)^3-1/2*B*c^2
/e^6/(e*x+d)^2-1/5*(A*b^2*e^3-6*A*b*c*d*e^2+6*A*c^2*d^2*e-3*B*b^2*d*e^2+12*B*b*c*d^2*e-10*B*c^2*d^3)/e^6/(e*x+
d)^5

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Maxima [A]  time = 1.09267, size = 485, normalized size = 1.9 \begin{align*} -\frac{210 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 4 \, A b^{2} d^{2} e^{3} + 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 70 \,{\left (5 \, B c^{2} d e^{4} + 2 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 35 \,{\left (10 \, B c^{2} d^{2} e^{3} + 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 21 \,{\left (10 \, B c^{2} d^{3} e^{2} + 4 \, A b^{2} e^{5} + 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 7 \,{\left (10 \, B c^{2} d^{4} e + 4 \, A b^{2} d e^{4} + 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{420 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/420*(210*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 4*A*b^2*d^2*e^3 + 4*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2*A*b*c)*d
^3*e^2 + 70*(5*B*c^2*d*e^4 + 2*(2*B*b*c + A*c^2)*e^5)*x^4 + 35*(10*B*c^2*d^2*e^3 + 4*(2*B*b*c + A*c^2)*d*e^4 +
3*(B*b^2 + 2*A*b*c)*e^5)*x^3 + 21*(10*B*c^2*d^3*e^2 + 4*A*b^2*e^5 + 4*(2*B*b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 +
2*A*b*c)*d*e^4)*x^2 + 7*(10*B*c^2*d^4*e + 4*A*b^2*d*e^4 + 4*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^
2*e^3)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d
^6*e^7*x + d^7*e^6)

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Fricas [A]  time = 1.69399, size = 771, normalized size = 3.02 \begin{align*} -\frac{210 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 4 \, A b^{2} d^{2} e^{3} + 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 70 \,{\left (5 \, B c^{2} d e^{4} + 2 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 35 \,{\left (10 \, B c^{2} d^{2} e^{3} + 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 21 \,{\left (10 \, B c^{2} d^{3} e^{2} + 4 \, A b^{2} e^{5} + 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 7 \,{\left (10 \, B c^{2} d^{4} e + 4 \, A b^{2} d e^{4} + 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{420 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/420*(210*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 4*A*b^2*d^2*e^3 + 4*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2*A*b*c)*d
^3*e^2 + 70*(5*B*c^2*d*e^4 + 2*(2*B*b*c + A*c^2)*e^5)*x^4 + 35*(10*B*c^2*d^2*e^3 + 4*(2*B*b*c + A*c^2)*d*e^4 +
3*(B*b^2 + 2*A*b*c)*e^5)*x^3 + 21*(10*B*c^2*d^3*e^2 + 4*A*b^2*e^5 + 4*(2*B*b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 +
2*A*b*c)*d*e^4)*x^2 + 7*(10*B*c^2*d^4*e + 4*A*b^2*d*e^4 + 4*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^
2*e^3)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d
^6*e^7*x + d^7*e^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.22808, size = 432, normalized size = 1.69 \begin{align*} -\frac{{\left (210 \, B c^{2} x^{5} e^{5} + 350 \, B c^{2} d x^{4} e^{4} + 350 \, B c^{2} d^{2} x^{3} e^{3} + 210 \, B c^{2} d^{3} x^{2} e^{2} + 70 \, B c^{2} d^{4} x e + 10 \, B c^{2} d^{5} + 280 \, B b c x^{4} e^{5} + 140 \, A c^{2} x^{4} e^{5} + 280 \, B b c d x^{3} e^{4} + 140 \, A c^{2} d x^{3} e^{4} + 168 \, B b c d^{2} x^{2} e^{3} + 84 \, A c^{2} d^{2} x^{2} e^{3} + 56 \, B b c d^{3} x e^{2} + 28 \, A c^{2} d^{3} x e^{2} + 8 \, B b c d^{4} e + 4 \, A c^{2} d^{4} e + 105 \, B b^{2} x^{3} e^{5} + 210 \, A b c x^{3} e^{5} + 63 \, B b^{2} d x^{2} e^{4} + 126 \, A b c d x^{2} e^{4} + 21 \, B b^{2} d^{2} x e^{3} + 42 \, A b c d^{2} x e^{3} + 3 \, B b^{2} d^{3} e^{2} + 6 \, A b c d^{3} e^{2} + 84 \, A b^{2} x^{2} e^{5} + 28 \, A b^{2} d x e^{4} + 4 \, A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}}{420 \,{\left (x e + d\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/420*(210*B*c^2*x^5*e^5 + 350*B*c^2*d*x^4*e^4 + 350*B*c^2*d^2*x^3*e^3 + 210*B*c^2*d^3*x^2*e^2 + 70*B*c^2*d^4
*x*e + 10*B*c^2*d^5 + 280*B*b*c*x^4*e^5 + 140*A*c^2*x^4*e^5 + 280*B*b*c*d*x^3*e^4 + 140*A*c^2*d*x^3*e^4 + 168*
B*b*c*d^2*x^2*e^3 + 84*A*c^2*d^2*x^2*e^3 + 56*B*b*c*d^3*x*e^2 + 28*A*c^2*d^3*x*e^2 + 8*B*b*c*d^4*e + 4*A*c^2*d
^4*e + 105*B*b^2*x^3*e^5 + 210*A*b*c*x^3*e^5 + 63*B*b^2*d*x^2*e^4 + 126*A*b*c*d*x^2*e^4 + 21*B*b^2*d^2*x*e^3 +
42*A*b*c*d^2*x*e^3 + 3*B*b^2*d^3*e^2 + 6*A*b*c*d^3*e^2 + 84*A*b^2*x^2*e^5 + 28*A*b^2*d*x*e^4 + 4*A*b^2*d^2*e^
3)*e^(-6)/(x*e + d)^7